The aggregated finite element method in [1]} was designed to solve this locking phenomenon and provide an accurate \({C}^0\)  Lagrangian finite element space on unfitted meshes that is easy to implement. The idea is to introduce the concept of ownership for ill-posed degrees of freedom. For each ill-posed global degree of freedom in \(V_h^{\textrm {ag}}\) , one assigns one of the aggregates in \(\mathcal {T}_h^{\mathrm {ag}}\)  containing this degree of freedom as the owner. The choice of the owner is arbitrary and does not affect the main properties of the method. We can define a map \(\mathcal {O}_{h}^{\mathrm {ipd} \rightarrow \mathrm {wpd}}\)  using the fact that each aggregate contains only one interior element. Given an ill-posed degree of freedom, its corresponding well-posed constraining degrees of freedom are those of the root element of the aggregate owner. We illustrate this construction in Figure REF . The purple elements are the ones that belong to more than one aggregate (the other ones are straightforward). For these degrees of freedom, the small rectangle points to the aggregate owner of the degree of freedom. With this construction, we can now constrain global ill-posed degrees of freedom using the same constraints as for discontinuous Galerkin methods (REF ).
